de Finetti Lattices and Magog Triangles

TL;DR

This paper explores the combinatorial structures related to de Finetti lattices, establishing bijections between linear extensions, Young tableaux, magog triangles, and alternating sign matrices, revealing new connections and symmetries.

Contribution

It introduces elementary bijections linking linear extensions of specific posets to Young tableaux and magog triangles, and uncovers involutions related to alternating sign matrices.

Findings

Bijection between linear extensions and shifted Young tableaux

Magog triangles correspond to minimal poset refinements

Row reversal in alternating sign matrices relates to an involution on gog triangles

Abstract

The order ideal $B_{n,2}$ of the Boolean lattice $B_n$ consists of all subsets of size at most $2$. Let $F_{n,2}$ denote the poset refinement of $B_{n,2}$ induced by the rules: $i < j$ implies $\{i \} \prec \{ j \}$ and $\{i,k \} \prec \{j,k\}$. We give an elementary bijection from the set $\mathcal{F}_{n,2}$ of linear extensions of $F_{n,2}$ to the set of shifted standard Young tableau of shape $(n, n-1, \ldots, 1)$, which are counted by the strict-sense ballot numbers. We find a more surprising result when considering the set $\mathcal{F}_{n,2}^{1}$ of minimal poset refinements in which each singleton is comparable with all of the doubletons. We show that $\mathcal{F}_{n,2}^{1}$ is in bijection with magog triangles, and therefore is equinumerous with alternating sign matrices. We adopt our proof techniques to show that row reversal of an alternating sign matrix corresponds to a natural involution on gog triangles.